A compound of formula `A_(2)B_(3)` has the hcp lattice. Which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atoms

Options

A. hcp lattice `-A, (2)/(3)` Tetrahedral voids-B
B. hcp lattice `-B, (1)/(3)` Tetrahdral voids -A
C. hcp lattice `-B, (2)/(3)` Tetrahedral voids -A
D. hcp lattive `-A, (1)/(3)` Tetrahdral voids-B

Answer

To solve the problem regarding the compound with the formula \( A_2B_3 \) that has a hexagonal close-packed (hcp) lattice, we need to determine which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atom. Here’s a step-by-step solution:

### Step 1: Understand the hcp structure
In a hexagonal close-packed structure, there are two layers of atoms (top and bottom hexagonal layers) and one layer of atoms in the middle. Each corner of the hexagon has an atom, and there are additional atoms at the face centers.

### Step 2: Determine the number of atoms in the hcp lattice
In an hcp lattice:
- There are 12 corner atoms (6 on the top hexagon and 6 on the bottom hexagon).
- Each corner atom contributes \( \frac{1}{6} \) of an atom to the unit cell.
- There are 2 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom.
- There are 3 atoms in the center of the unit cell.

Calculating the total number of atoms:
- Contribution from corners: \( 12 \times \frac{1}{6} = 2 \)
- Contribution from face centers: \( 2 \times \frac{1}{2} = 1 \)
- Contribution from the center: \( 3 \)

Total effective number of atoms in the hcp unit cell = \( 2 + 1 + 3 = 6 \).

### Step 3: Assign atoms to the lattice
Given the formula \( A_2B_3 \), we can assume that:
- If \( B \) forms the hcp lattice, then \( B \) occupies the 6 positions in the lattice.
- Therefore, \( B \) occupies the hcp lattice.

### Step 4: Calculate the number of tetrahedral voids
In an hcp structure, the number of tetrahedral voids is double the number of atoms present in the lattice. Since we have 6 atoms of \( B \):
- Number of tetrahedral voids = \( 2 \times 6 = 12 \).

### Step 5: Determine the fraction of tetrahedral voids occupied by \( A \)
We have \( A_2B_3 \), which means there are 2 atoms of \( A \). To find out how many tetrahedral voids are occupied by \( A \):
- If \( A \) occupies 4 tetrahedral voids, then the fraction of tetrahedral voids occupied by \( A \) is:
\[ \text{Fraction occupied} = \frac{\text{Number of A atoms}}{\text{Total tetrahedral voids}} = \frac{4}{12} = \frac{1}{3}. \]

### Conclusion
- The atom \( B \) forms the hcp lattice.
- The fraction of tetrahedral voids occupied by \( A \) is \( \frac{1}{3} \).